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  • The Jacobson radical and Nakayama's lemma

    Proposition Let a be an ideal of a ring A, and let S=1+a. Show that S1a is contained in the Jacobson radical of S1A. Use this result and Nakayama’s lemma to give a proof of (2.5)[Atiyah] which does not depend on determinants. Solution 0a,...


  • If f has a zero of multiplicity m at a, then 1/f has a pole of order m at a

    Proposition Suppose that f has a zero of multiplicity m at a. Explain why 1/f has a pole of order m at a. Solution f has a zero of multiplicity m at a. By Theorem 8.14[A first course in complex analysis], this means that there exists a holomorphic function g...


  • Poles and removable singularities

    Proposition Find the poles or removable singularities of the following functions and determine their orders: (z2+1)3(z1)4. zcot(z). z5sinz. z1exp(z). 11exp(z). Solution 1 The order of the pole at i is 3. The order of the pole at 1 is 4. 2...


  • If f has an essential singularity at z0, then so does 1/f

    Proposition Show that if f has an essential singularity at z0 then 1f also has an essential singularity at z0. Solution Suppose 1/f does not have an essential singularity at z0. Since f has an isolated singularity at z0, 1/f has an isolated singularity at z0. By Proposition 9.5[a first...


  • Cross products and determinants

    Proposition If w1,,wn1Rn, show that |w1××wn1|=det(gij), where gij=wi,wj. Solution Suppose w1,,wn1 are linearly dependent. Then w1××wn1=0 because $\ev{w_1 \times...